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ON THE MOTION  OF PENDULUMS.                          43
The equivalence of the expressions (87) and (89) having been ascertained, in order to find the relations between C, D and C", D", it will be sufficient to write down the two leading terms in (87) and (89), and equate the results. We thus get
",    D = 7rD" ............ (90).
There remains the more difficult step of finding the relation between D' and C", D". For this purpose let us seek the ultimate value of the second member of equation (89) when r increases indefinitely. In the first place we may observe that if L, 1' be two imaginary quantities having their real parts positive, if the real part of H be greater than that of H', and if r be supposed to increase indefinitely, Gr will ultimately be incomparably greater than eov, or even than log r. ear, or, to speak more precisely, the modulus of the former expression will ultimately be incomparably greater than the modulus of either of the latter. Hence, in finding the ultimate value of the expression for Fs(r) in (89), we may replace the limits 0 and ^TT of w by 0 and a>^ where 1 is a positive quantity as small as we please, which we may suppose to vanish after r has become infinite. We may also, for the same reason, omit the second of the exponentials. Let cos co = 1  X, so that
- X2 then the limits of X will be 0 and \, where \ = 1  cos &>r    Since
log ( 1  ^ ) ultimately vanishes, and 1 + 7 +  .  becomes ultimately \        A/                                            4
1, we get from (89)
limit of Ffr) = emr x limit of f *\C" + D" log 2Xr)
due to Euler, will be found in the second volume of the Cambridge Mathematical Journal, p. 282, or in Gregory's Examples, p. 484.
* The word limit is here used in the sense in which /(r) may be called the limit of 0 (r) when the ratio of 0 (r) to /(r) is ultimately a ratio of equality, though f(r) and 0 (r) may vanish or become infinite together, in which case the limit of 0(r), according to the usual sense of the word limit, would be said to be zero or infinity.